# Properties

 Label 605.2.m.g Level $605$ Weight $2$ Character orbit 605.m Analytic conductor $4.831$ Analytic rank $0$ Dimension $160$ CM no Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.m (of order $$20$$, degree $$8$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$160$$ Relative dimension: $$20$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$160q - 8q^{3} + 4q^{5} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$160q - 8q^{3} + 4q^{5} - 128q^{12} - 32q^{15} + 24q^{16} - 32q^{20} - 96q^{23} + 52q^{25} + 8q^{26} - 32q^{27} - 16q^{31} + 40q^{36} - 36q^{37} - 32q^{38} - 32q^{42} + 256q^{45} - 32q^{47} + 16q^{48} + 68q^{53} + 320q^{56} - 132q^{58} + 64q^{60} - 352q^{67} + 8q^{70} + 16q^{75} - 992q^{78} + 164q^{80} + 40q^{81} - 100q^{82} + 80q^{86} + 96q^{91} + 56q^{92} - 24q^{93} + 68q^{97} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
112.1 −1.23116 2.41628i −0.183968 + 1.16153i −3.14709 + 4.33160i −0.956130 + 2.02134i 3.03306 0.985502i −0.535369 + 0.0847942i 8.98397 + 1.42292i 1.53787 + 0.499685i 6.06127 0.178306i
112.2 −1.05904 2.07848i −0.318551 + 2.01125i −2.02296 + 2.78437i 2.18295 + 0.484484i 4.51772 1.46790i 4.54674 0.720133i 3.32162 + 0.526094i −1.09050 0.354324i −1.30484 5.05032i
112.3 −1.05076 2.06223i 0.159862 1.00933i −1.97313 + 2.71579i −1.98733 + 1.02495i −2.24944 + 0.730888i 2.99845 0.474908i 3.10187 + 0.491288i 1.85999 + 0.604346i 4.20189 + 3.02136i
112.4 −0.890440 1.74759i 0.459138 2.89888i −1.08561 + 1.49421i 0.260122 2.22089i −5.47488 + 1.77890i −2.07637 + 0.328864i −0.296505 0.0469618i −5.33954 1.73492i −4.11281 + 1.52298i
112.5 −0.832753 1.63437i −0.488992 + 3.08737i −0.802115 + 1.10402i −1.82276 + 1.29520i 5.45312 1.77183i 0.491081 0.0777795i −1.15109 0.182315i −6.43959 2.09235i 3.63475 + 1.90048i
112.6 −0.679174 1.33295i −0.340390 + 2.14914i −0.139917 + 0.192579i 0.485485 2.18273i 3.09589 1.00591i −2.32674 + 0.368519i −2.60345 0.412346i −1.64977 0.536043i −3.23920 + 0.835322i
112.7 −0.544755 1.06914i 0.0572260 0.361310i 0.329265 0.453195i 2.07465 0.834158i −0.417466 + 0.135643i 1.48586 0.235337i −3.03420 0.480570i 2.72590 + 0.885698i −2.02201 1.76368i
112.8 −0.323023 0.633967i 0.440005 2.77808i 0.877999 1.20846i −2.10640 0.750378i −1.90335 + 0.618434i 3.39829 0.538236i −2.45526 0.388874i −4.67097 1.51769i 0.204700 + 1.57778i
112.9 −0.280420 0.550355i −0.235077 + 1.48421i 0.951315 1.30937i −2.19624 0.420141i 0.882765 0.286828i −1.51988 + 0.240726i −2.20753 0.349639i 0.705537 + 0.229243i 0.384643 + 1.32653i
112.10 −0.120168 0.235842i 0.00828396 0.0523028i 1.13439 1.56135i 1.85452 + 1.24930i −0.0133307 + 0.00433140i −3.21766 + 0.509627i −1.02742 0.162727i 2.85050 + 0.926184i 0.0717827 0.587500i
112.11 0.120168 + 0.235842i 0.00828396 0.0523028i 1.13439 1.56135i 1.85452 + 1.24930i 0.0133307 0.00433140i 3.21766 0.509627i 1.02742 + 0.162727i 2.85050 + 0.926184i −0.0717827 + 0.587500i
112.12 0.280420 + 0.550355i −0.235077 + 1.48421i 0.951315 1.30937i −2.19624 0.420141i −0.882765 + 0.286828i 1.51988 0.240726i 2.20753 + 0.349639i 0.705537 + 0.229243i −0.384643 1.32653i
112.13 0.323023 + 0.633967i 0.440005 2.77808i 0.877999 1.20846i −2.10640 0.750378i 1.90335 0.618434i −3.39829 + 0.538236i 2.45526 + 0.388874i −4.67097 1.51769i −0.204700 1.57778i
112.14 0.544755 + 1.06914i 0.0572260 0.361310i 0.329265 0.453195i 2.07465 0.834158i 0.417466 0.135643i −1.48586 + 0.235337i 3.03420 + 0.480570i 2.72590 + 0.885698i 2.02201 + 1.76368i
112.15 0.679174 + 1.33295i −0.340390 + 2.14914i −0.139917 + 0.192579i 0.485485 2.18273i −3.09589 + 1.00591i 2.32674 0.368519i 2.60345 + 0.412346i −1.64977 0.536043i 3.23920 0.835322i
112.16 0.832753 + 1.63437i −0.488992 + 3.08737i −0.802115 + 1.10402i −1.82276 + 1.29520i −5.45312 + 1.77183i −0.491081 + 0.0777795i 1.15109 + 0.182315i −6.43959 2.09235i −3.63475 1.90048i
112.17 0.890440 + 1.74759i 0.459138 2.89888i −1.08561 + 1.49421i 0.260122 2.22089i 5.47488 1.77890i 2.07637 0.328864i 0.296505 + 0.0469618i −5.33954 1.73492i 4.11281 1.52298i
112.18 1.05076 + 2.06223i 0.159862 1.00933i −1.97313 + 2.71579i −1.98733 + 1.02495i 2.24944 0.730888i −2.99845 + 0.474908i −3.10187 0.491288i 1.85999 + 0.604346i −4.20189 3.02136i
112.19 1.05904 + 2.07848i −0.318551 + 2.01125i −2.02296 + 2.78437i 2.18295 + 0.484484i −4.51772 + 1.46790i −4.54674 + 0.720133i −3.32162 0.526094i −1.09050 0.354324i 1.30484 + 5.05032i
112.20 1.23116 + 2.41628i −0.183968 + 1.16153i −3.14709 + 4.33160i −0.956130 + 2.02134i −3.03306 + 0.985502i 0.535369 0.0847942i −8.98397 1.42292i 1.53787 + 0.499685i −6.06127 + 0.178306i
See next 80 embeddings (of 160 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 602.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
55.e even 4 1 inner
55.k odd 20 3 inner
55.l even 20 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.m.g 160
5.c odd 4 1 inner 605.2.m.g 160
11.b odd 2 1 inner 605.2.m.g 160
11.c even 5 1 605.2.e.c 40
11.c even 5 3 inner 605.2.m.g 160
11.d odd 10 1 605.2.e.c 40
11.d odd 10 3 inner 605.2.m.g 160
55.e even 4 1 inner 605.2.m.g 160
55.k odd 20 1 605.2.e.c 40
55.k odd 20 3 inner 605.2.m.g 160
55.l even 20 1 605.2.e.c 40
55.l even 20 3 inner 605.2.m.g 160

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.e.c 40 11.c even 5 1
605.2.e.c 40 11.d odd 10 1
605.2.e.c 40 55.k odd 20 1
605.2.e.c 40 55.l even 20 1
605.2.m.g 160 1.a even 1 1 trivial
605.2.m.g 160 5.c odd 4 1 inner
605.2.m.g 160 11.b odd 2 1 inner
605.2.m.g 160 11.c even 5 3 inner
605.2.m.g 160 11.d odd 10 3 inner
605.2.m.g 160 55.e even 4 1 inner
605.2.m.g 160 55.k odd 20 3 inner
605.2.m.g 160 55.l even 20 3 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$54\!\cdots\!46$$$$T_{2}^{132} +$$$$18\!\cdots\!46$$$$T_{2}^{128} -$$$$55\!\cdots\!82$$$$T_{2}^{124} +$$$$14\!\cdots\!23$$$$T_{2}^{120} -$$$$34\!\cdots\!76$$$$T_{2}^{116} +$$$$72\!\cdots\!85$$$$T_{2}^{112} -$$$$13\!\cdots\!00$$$$T_{2}^{108} +$$$$20\!\cdots\!05$$$$T_{2}^{104} -$$$$29\!\cdots\!82$$$$T_{2}^{100} +$$$$37\!\cdots\!44$$$$T_{2}^{96} -$$$$39\!\cdots\!10$$$$T_{2}^{92} +$$$$37\!\cdots\!19$$$$T_{2}^{88} -$$$$31\!\cdots\!84$$$$T_{2}^{84} +$$$$21\!\cdots\!13$$$$T_{2}^{80} -$$$$11\!\cdots\!56$$$$T_{2}^{76} +$$$$54\!\cdots\!99$$$$T_{2}^{72} -$$$$22\!\cdots\!46$$$$T_{2}^{68} +$$$$71\!\cdots\!48$$$$T_{2}^{64} -$$$$15\!\cdots\!58$$$$T_{2}^{60} +$$$$29\!\cdots\!57$$$$T_{2}^{56} -$$$$49\!\cdots\!80$$$$T_{2}^{52} +$$$$59\!\cdots\!09$$$$T_{2}^{48} -$$$$22\!\cdots\!76$$$$T_{2}^{44} +$$$$67\!\cdots\!27$$$$T_{2}^{40} -$$$$18\!\cdots\!46$$$$T_{2}^{36} +$$$$44\!\cdots\!30$$$$T_{2}^{32} -$$$$62\!\cdots\!82$$$$T_{2}^{28} +$$$$82\!\cdots\!89$$$$T_{2}^{24} -$$$$96\!\cdots\!36$$$$T_{2}^{20} +$$$$83\!\cdots\!79$$$$T_{2}^{16} -$$$$41\!\cdots\!68$$$$T_{2}^{12} +$$$$20\!\cdots\!43$$$$T_{2}^{8} -$$$$98\!\cdots\!66$$$$T_{2}^{4} +$$$$45\!\cdots\!61$$">$$T_{2}^{160} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$.